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Estimation of integrals with respect to infinite measures using regenerative sequences

Part of:Limit theoremsProbabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 March 2016

Krishna B. Athreya and
Vivekananda Roy
Krishna B. Athreya*
Affiliation:
Iowa State University
Vivekananda Roy*
Affiliation:
Iowa State University
*
∗ Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
∗∗ Postal address: Department of Statistics, Iowa State University, Ames, IA 50011, USA. Email address: vroy@iastate.edu
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Abstract

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Let f be an integrable function on an infinite measure space (S, , π). We show that if a regenerative sequence {Xn }n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫S f dπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and integrals with respect to absolutely continuous measures on ℝd using a simple symmetric random walk on ℤ.

Keywords

Markov chainMonte Carloimproper targetrandom walkregenerative sequence

MSC classification

Primary:65C05: Monte Carlo methods
Secondary:60F05: Central limit and other weak theorems

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015, pp. 1133 - 1145
Copyright
Copyright © Applied Probability Trust 2015 

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